Reconnect 04 Introduction to Integer Programming

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Reconnect 04Introduction to Integer Programming

• Cynthia Phillips, Sandia National Laboratories

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Integer programming (IP)

• Min
• Subject to
• Can also have inequalities in either direction
  (slack variables)
• If , then this is a
  mixed-integer program (MIP)
• Linear programming (LP) has no integrality
  constraints (in P)
• IP (easily) expresses any NP-complete problem

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Terminology

• In this context programming means making
  decisions.
• Leading terms say what kind
• (Pure) Integer programming all integer decisions
• Linear programming
• Quadratic programming quadratic objective
  function
• Nonlinear programming nonlinear constraints
• Stochastic programming finite probability
  distribution of scenarios
• Came from operations research (practical
  optimization discipline)
• Computer programming (by someone) is required to
  solve these.

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Decisions

• The IPs Ive encountered in practice involve
  either
• Allocation of scarce resources
• Study of a natural system
• Computational biology
• Mathematics
• Maybe during or after this course, you can add to
  the list

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Integer Variables

• Use (binary variables) to model
• Yes/no decisions
• Disjunctions
• Logical conditions
• Piecewise linear functions (this not covered in
  this lecture)

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General Integer Variables

• Use general integer variables to choose a number
  of indivisible objects such as the number of
  planes to produce
• Integer range should be small (e.g. 1-10)
• Computational tractability
• Larger ranges may be well approximated by
  rational variables (number of bags of potato
  chips to produce)

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Example Binary Knapsack

• Given set of objects 1..n
• total weight W, item weight/size wi, value vi

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Example Shortest-Path Network Interdiction

• Delay an adversary moving through a network.
• Adversary moves start?target along a shortest
  path (in worst case)
• Path length sum of edge lengths. Measure of
  time, exposure, etc.

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Target
3
1
Start
2
2
5
1
4
Shortest Path Length 8
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Example Shortest-Path Network Interdiction

• Defender blocks the intruder by paying to
  increase edge lengths.
• Goal Maximize the resulting shortest path.

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Target
3
1
Start
2
2
1
5
2
1
4
Shortest Path Length 11
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Path Interdiction Mixed-Integer Program

• Graph G (V,E)
• Edge lengths for edge (u,v)
• Can increase length of (u,v) by ?uv at cost cuv
• Budget B
• Variables
• du shortest distance from start s to node
  u

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Path Interdiction Integer Program

• Objective maximize the shortest path to the
  target
• maximize dt
• Subject to
• Path to the start has length 0
• ds 0
• Calculate a shortest path length
• Respect the budget

u
?uv
v
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Modeling Dependent Decisions

• Suppose x,y are two binary variables that
  represent a decision (where 1 means yes and 0
  means no)
• The constraint allows x to be
  yes only if y is yes

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Example Unconstrained Facility Location

• Given potential facility locations, n customers
  to be served
• cj cost to build facility j
• hij cost to meet all of customer is demand
  from facility j
• Sometimes its OK to satisfy customers from
  multiple facilities

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Formulation is really important in practice

• Unconstrained facility location
• Could sum constraints
  over all customers i to get
• Recall n is the number of customers.
• Still requires a facility is built before use
  (IPs are equivalent at optimality)
• But, for 40 customers, 40 facilities, random
  costs
• First formulation solves in 2 seconds
• Second formulation solves in 53,121 seconds
  (14.75 hours)

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What makes one formulation so much better?

• Understanding this fully is an open problem.
• Some performance differences can be explained by
  the way IPs are solved in practice by
  branch-and-bound-like algorithms the LP
  relaxation

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Recall Integer Programming (IP)

• Min
• Subject to

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Linear programming (LP) relaxation of an IP

• Min
• Subject to
• LP can be solved efficiently (in theory and
  practice)
• Relaxation removing constraints
• All feasible IP solutions are feasible
• LP gives a lower bound

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Linear Programming Geometry

• The solutions to a single inequality
  form a half space (in
  n-dimensional space)

feasible
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Linear Programming Geometry

• Intersection of all the linear (in)equalities
  form a convex polytope
• For simplicity, well always assume polytope is
  bounded

feasible
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IP Geometry

• Feasible integer points form a lattice inside the
  LP polytope

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IP Geometry

• The convex hull of this lattice forms the integer
  polytope

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IP/LP Geometry

• A good formulation keeps this region small
• Every node for which the LP bound is lower than
  the integer optimal must be processed (e.g.
  expanded)

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IP/LP Geometry

• A good formulation keeps this region small
• One measure of this is the Integrality Gap
• Integrality gap maxinstances I(IP (I))/(LP(I))

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Unconstrained Facility Location Revisited

• Given potential facility locations, customers to
  be served
• cj cost to build facility j
• hij cost to meet all of customer is demand
  from facility j

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How the weaker LP cheats

• Using
• Allows the LP to completely satisfy customer i
  with facility j (yij 1) even with xj 1/n.
• With these constraints
• If xi 1/n, then yij lt 1/n

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Cant we just round the LP Solution?

• Not generally feasible
• If (miraculously) it is feasible, its not
  generally good

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Example Maximum Independent Set

• Find a maximum-size set of vertices that have no
  edges between any pair

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Example Maximum Independent Set
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Example Maximum Independent Set

• The zero-information solution (vi .5 for all i)
  is feasible and its optimal if the optimal MIS
  has size at most V/2.
• Rounding everything (up) is infeasible.

1/2
1/2
1/2
1/2
1/2
1/2
1/2
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Cant we project the lattice onto the objective
gradient?

• Hard to find a feasible solution to project
  (NP-complete!)
• Make the objective a constraint and do binary
  search
• This is a lot harder in n dimensions than it
  looks like in 2

gradient
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Perfect formulations

• Sometimes solving an LP is guaranteed to give an
  integer solution
• All polytope corners have integer coefficients
  (naturally integer)
• Sometimes only for specific objectives (e.g.
  )

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Perfect Formulation Example Minimum Cut
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1
4
Capacity ue
27
2
s
5
5
9
3
3
1
20
t
8
2
5
2

• Special nodes s and t
• Each edge e has capacity ue. Set of edges S has
  capacity
• Partition vertex set V into S,T where
• A cut is the edges (u,v) such that
• Find a cut with minimum capacity

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Perfect Formulation Example Minimum Cut IP

• Helper variables ye 1 if e is in the cut and 0
  otherwise
• The y variables will be integral if the v
  variables are.

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Total Unimodularity

• The minimum cut matrix (possibly with slack
  variables) is totally unimodular (TU) all
  subdeterminants (including the matrix entries)
  have value 0, 1, or -1.
• All corner solutions x satisfy Axb
• By Kramers rule x will be integral
• Network matrices (adjacency matrices of graphs)
  are TU.
• Nemhauser and Wolsey (Integer and Combinatorial
  Optimization, Wiley, 1988) give some sufficient
  conditions for a matrix to be TU.
• Note if a matrix is TU, there is always an
  efficient combinatorial algorithm to solve the
  problem (not necessarily obvious)

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Total Unimodularity is Fragile

• Example Network Interdiction
• Expend a limited budget to maximally damage the
  transport capacity of a network

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Network Flow
11/12
1
4
11/27
0/5
9/9
2/2
5/5
s
3
1/1
8/20
t
3/3
2
5
2/8
2/2

• Source(s) s, sink (consumers) t
• Capacity (bottom number)
• Flow (top number)
• Maximize flow from s to t obeying
• Capacity constraints on edges
• Conservation constraints on all nodes other than
  s,t

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Network Interdiction
11/12
1
4
11/27
0/5
9/9
2/2
5/5
s
3
1/1
8/20
t
3/3
2
5
2/8
2/2

• Each edge e now has a destruction cost de (cost
  to remove e assume linear)
• Budget B
• Expend at most B removing (pieces of) edges in
  the network so resulting max flow is minimized

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Network Interdiction

• By LP duality (well see later)
• value of max flow value of min cut
• So
• minattacks max flow minattacks min cut
• Pay to knock out transport capacity from s to t

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1
4
27
5
s
5
9
2
3
3
1
20
t
8
2
5
2
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A Mixed Integer Program for Network Inhibition
0
1
t
s

• Based on min-cut LP
• Find best cut to attack
• Decision variables place vertices on the s or t
  side as before
• All edges going across the cut must be destroyed
  (consume budget) or contribute to residual cut
  capacity

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Network Inhibition IP

• Helper variables ye percent of an edge in cut
  that is not removed
• ze percent of an
  edge in the cut that is destroyed

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Total Unimodularity is Fragile

• The matrix is still TU without the budget
  constraint
• Adding the budget constraint makes the problem
  strongly NP-complete
• No known polynomial-time approximation algorithms
• Still has some very nice structure that gives a
  pseudo-approximation
• Pseudo-approximation might give a superoptimal
  solution that slightly exceeds the budget or it
  could give a true approximation

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Modeling Sets

• Given a set T,
• means select at least
  1 element of T
• Making sure at least one local warehouse has
  inventory for each customer
• means select at most 1
  element of T
• Conflicts (e.g. modeled by a maximum independent
  set problem)
• Resource constraints
• means select exactly 1
  element of T
• Time indexed scheduling variables xjt, schedule
  job j at time t. This picks a single time for job
  j.

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Modeling Disjunctive Constraints

• Let be two
  constraints with nonnegative coefficients
• To force satisfaction of at least one of these
  constraints

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Modeling Disjunctive Constraints - General Number

• Let be m
  constraints with nonnegative coefficients
• To force satisfaction of at least k of these
  constraints

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Modeling a Restricted Set of Values

• Variable x can take on only values in
• Frequently the vi are sorted
• Example capacity of an airplane assigned to a
  flight
• The yis are a special ordered set.

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Some simple logical constraints

• Want (logical or)
• Suffices if there is pressure in the objective
  function to keep y low.
• Saw this in minimum cut
• Similarly if we want
  (logical and)
• Suffices if there is pressure in the objective
  function to keep y high.

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Example Protein Structure Comparison

• 2 nonadjacent amino acids share an edge if
  theyre physically close when folded

Contact Map
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Example Protein Structure Comparison

• 2 nonadjacent amino acids share an edge if
  theyre physically close folded
• Noncrossing alignment of two proteins to maximize
  shared contacts
• Measure of similarity

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Protein Structure Comparison

• Variables xij 1 if amino acid in position i of
  the top protein is matched to amino acid in
  position j of the bottom protein, 0 otherwise
• Helper variables

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Non-crossing alignment

• For any pair of edges, we can tell if they cross
• if the pair is forbidden (simply dont create
  this variable).
• There are more clever ways to do this (e.g. using
  Ramsey theory). See what you can come up with.

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Protein Structure Comparison

• Only consider yijkl if this is a shared contact
  ((i,k) a contact, (j,l) a contact)

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MIP Applications (Small Sample)

• Logistics
• Capacity planning, scheduling, workforce
  planning, military spares management
• Infrastructure/network security
• Vulnerability analysis, reinforcement,
  reliability, design, integrity of physical
  transport media
• Sensor placement (water systems, roadways)
• Waste remediation
• Vehicle routing, fleet planning
• Bioinformatics protein structure
  prediction/comparison, drug docking
• VLSI, robot design
• Tools for high-performance computing (scheduling,
  node allocation, domain decomposition, meshing)

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Solving Integer Programs

• NP-hard
• Many special cases have efficient solutions or
  provably-good approximation bounds
• Need time to explore structure
• General IPs can be hard due to size and/or
  structure
• (Sufficiently) optimal solution is important
• When lives or big at stake
• For rigorous benchmarking of heuristic/approximati
  on methods
• To gain structural insight for better
  algorithms/proofs.